Section 1.3
More on Functions and
Their Graphs
Math 112
Section 1.3
A function is said to be increasing on an interval if, for
all a and b in the interval, a > b implies f(a) > f(b).
Or simply put, increasing means up, decreasing means
down, and constant means level.
Increasing: (4 , 1)  (4 , )
Decreasing: ( , 4)
Constant: (1 , 4)
Relative Minimum: 2
The intervals describing
where functions increase,
decrease, or are constant,
use x-coordinates and
not the y-coordinates.
Find where the graph is increasing?
Where is it decreasing? Where is it
constant?
Example
y






x



























Example
y
Find where the graph is increasing? Where
is it decreasing? Where is it constant?






x























Relative Maxima
And
Relative Minima
Where are the relative minimums?
Where are the relative maximums?
Example
Why are the maximums and minimums
called relative or local?
y






x


























Even and Odd Functions
and Symmetry
Math 112
A graph is symmetric with respect to the
y-axis if for any point (x , y) on the graph,
the point (x , y) is also on the graph.
(x , y)
(x , y)
Math 112
Section 1.7
A graph is symmetric with respect to the
origin if for any point (x , y) on the graph,
the point (x , y) is also on the graph.
(x , y)
(x , y)
Math 112
Section 1.7
If the graph of a function f is symmetric with
respect to the y-axis, (that is, f(x) = f(x)),
then f(x) is an even function.
If the graph of a function f is symmetric with
respect to the origin, (that is, f(x) = f(x)),
then f(x) is an odd function.
Math 112
Even Function
(x , y)
Odd Function
(x , y)
(x , y)
(x , y)
Example
Is this an even or odd function?
y






x


























Example
Is this an even or odd function?
y






x


























Example
Is this an even or odd function?
y






x


























Math 112
Even Function
f(x) = 3x2  10
f(x) = 3(x)2 
10
= 3x2  10
= f(x)
f(x) = f(x)
Odd Function
f(x) = 6x3  4x
f(x) = 6(x)3 
4(x)
= 6x3 + 4x
= (6x3  4x)
= f(x)
f(x) = f(x)
Piecewise Functions
Math 112
A piecewise defined function is a function
defined by different formulas for different
parts of the domain.
for x   4
3

f(x)  2x  1 for  4  x  4
x 2
for x  4

f(4) = 3
f(2) = 2(2) + 1 = 5
f(4) = 42 = 16
Math 112
for x   4
3

f(x)  2x  1 for  4  x  4
x 2
for x  4

21
21
A function that is defined by two or more equations over
a specified domain is called a piecewise function. Many
cellular phone plans can be represented with piecewise
functions. See the piecewise function below:
A cellular phone company offers the following plan:
$20 per month buys 60 minutes
Additional time costs $0.40 per minute.
C t  
20
if 0  t  60
20  0.40(t  60) if t>60
Example
C t  
20
if 0  t  60
20  0.40(t  60) if t>60
Find and interpret each of the following.
C  45 
C  60 
C  90 
Example
Graph the following piecewise function.
3
if -  x  3
2 x  3 if x>3
f  x 
y






x
























Functions and
Difference Quotients
See next slide.
f(x+h)-f(x)
2
Find
for f(x)=x  2 x  5
h
First find f(x+h)
f(x+h)=(x+h)  2(x+h)-5
2
x  2hx  h  2 x  2h  5
2
2
Continued on the next slide.
f(x+h)-f(x)
Find
for f(x)=x 2  2 x  5
h
Use f(x+h) from the previous slide
f(x+h)-f(x)
Second find
h
2
2
2
x

2
hx

h

2
x

2
h

5

x
 2 x  5

f(x+h)-f(x)

h
h
x 2  2hx  h 2  2 x  2h  5  x 2  2 x  5
h
2hx  h 2  2h
h
h  2x  h  2
h
2x+h-2
Example
Find and simplify the expressions if f ( x)  2 x  1
Find f(x+h)
f(x+h)-f(x)
Find
, h 0
h
Example
2
f
(
x
)

x
4
Find and simplify the expressions if
f(x+h)-f(x)
Find f(x+h)
Find
, h 0
h
Example
2
f
(
x
)

x
 2x 1
Find and simplify the expressions if
Find f(x+h)
f(x+h)-f(x)
Find
, h 0
h
y

There is a relative minimum at x=?


























Find the difference quotient for f(x)=3x 2 .
Evaluate the following piecewise function at f(-1)
2x+1 if x<-1
f(x)=
-2 if -1  x  1
x-3 if x>1